Question

the amount of money irone earns for walking dogs is shown in the table
irones earnings
dogs walked: 2, 4, 6, 8
money earned: $10.50, $21.00, $31.50, $42.00
how much money does she earn for walking 15 dogs?
write the equation in the form of y = kx that represents this table.
10 the table below shows the costs of buying concert tickets online
cost of tickets
number of tickets: 1, 2, 3, 4
cost ($): $30, $50, $70, $90
part a
graph a line that represents the relationship shown in the table.
graph: y-axis labeled cost ($), x-axis labeled number of tickets, with grid from 0-100 (y) and 0-5 (x)
part b
does the graph show a proportional relationship?
circle the correct answers to complete the sentences.
the line does / does not pass through all four points and the origin
therefore, the relationship in the graph is / is not a proportional relationship.

Explanation:

First Problem (Irene's Earnings)

Step 1: Find the constant of proportionality \( k \)

We use the formula \( y = kx \), where \( y \) is money earned and \( x \) is dogs walked. Using the first pair \((2, 10.50)\):
\( k=\frac{y}{x}=\frac{10.50}{2} = 5.25 \)

Step 2: Write the equation

The equation is \( y = 5.25x \).

Step 3: Calculate earnings for 15 dogs

Substitute \( x = 15 \) into \( y = 5.25x \):
\( y = 5.25\times15 = 78.75 \)

Second Problem (Concert Tickets)

Part A: Graphing the Line

• Plot the points \((1, 30)\), \((2, 50)\), \((3, 70)\), \((4, 90)\) on the coordinate plane (x-axis: Number of Tickets, y-axis: Cost).
• Draw a straight line through these points.

Part B: Proportional Relationship

• Check if the line passes through the origin \((0,0)\). For \( x = 0 \), \( y \) should be \( 0 \) for a proportional relationship. Let's check the ratio \( \frac{y}{x} \) for each point:
• \( \frac{30}{1}=30 \), \( \frac{50}{2}=25 \), \( \frac{70}{3}\approx23.33 \), \( \frac{90}{4}=22.5 \). The ratios are not constant, so the line does not pass through the origin (or have a constant \( k \)).
• Thus, the line does not pass through all four points and the origin, and the relationship is not proportional.

Answer:

s:

• Irene earns \(\boldsymbol{\$78.75}\) for walking 15 dogs. The equation is \(\boldsymbol{y = 5.25x}\).
• Concert Tickets Part A: Graph with points \((1,30)\), \((2,50)\), \((3,70)\), \((4,90)\) connected.
• Concert Tickets Part B: The line \(\boldsymbol{\text{does not}}\) pass through all four points and the origin. The relationship \(\boldsymbol{\text{is not}}\) proportional.